Download Intermediate Algebra Section 4.1 – Systems of Linear Equations in

Intermediate Algebra
Section 4.1 – Systems of Linear Equations in Two
Variables
A system of equations involves more than one variable and more than
one equation. In this section we will focus on systems containing
two equations and two variables. For example, a system of two
equations in x and y is
x − y = −4
x + 2y = 5
A solution is an ordered pair (x, y ) that satisfies both equations.
Since a solution of a system of two equations in two variables is a
solution common to both equations, it is also a point common to the
graphs of both equations. This is the point where the two graphs
intersect.
A system of equations that has at least one solution is said to be a
consistent system. A system that has no solution is said to be
inconsistent. A system that has infinitely many solutions is said to be
coincident.
Section 4.1 – Systems of Linear Equations in Two Variables
page 2
Determine whether the ordered pair (4, −7) is a solution
− x + y = − 11
to the system: 
.
2 x − 5 y = 43
Example:
Solving a system of linear equations by graphing is not usually a very
effective way to solve systems. There are other methods that should
be used because they are accurate. We will look at a couple of other
methods for solving linear equations. In this section, we will solve
systems using the substitution method.
Solving a System of Linear Equations by the Substitution
Method.
1. Solve one of the equations for one of its variables.
2. Substitute the expression for the variable found in step 1 into
the other equation.
3. Solve the equation from step 2 to find the value of one
variable.
4. Substitute the value found in step 3 in any equation containing
both variables to find the value of the other variable.
5. Check the proposed solution in the original equation.
Section 4.1 – Systems of Linear Equations in Two Variables
page 3
Example:
Solve the following systems first by graphing and then
y
using substitution.
a)
5
y = 2x + 3
4
5y − 7x = 18
3
2
1
-5
-4
-3
-2
-1
1
2
3
4
x
5
-1
-2
-3
-4
-5
y
b)
x + y = −3
5
2x + 2y = 6
4
3
2
1
-5
-4
-3
-2
-1
1
-1
-2
-3
-4
-5
2
3
4
5
x
Section 4.1 – Systems of Linear Equations in Two Variables
page 4
A consistent system of linear equations has at least one solution, and
an inconsistent system has no solution. You can see from the graph
in the previous example that an inconsistent system consists of lines
that have the same slope and different y -intercept. Algebraically,
you will know that you have an inconsistent system if the variables
are all eliminated and you are left with a false statement, as was the
case in the previous example. There are two types of consistent
linear systems. Linear systems whose graphs intercept at exactly one
point, which means the lines have different slopes, and linear systems
that intersect at infinitely many points, which means the lines have
the same slope and the same y -intercept.
Another method we can use to solve systems of linear equations is
called the elimination method. The elimination method is based on
the addition property of equality, adding equal quantities to both
sides of an equation does not change the solution of the equation.
When using the elimination method, we want to make it so that the
coefficient of one of the variables are opposites, so that when we add
the equations one of the variables is eliminated.
Solving a System of Two Linear Equations by the Elimination
Method
1. Rewrite each equation in standard form Ax + By = C .
2. If necessary, multiply one or both equations by a nonzero
number so that the coefficients of a chosen variable in the
system are opposites.
3. Add the equations.
4. Find the value of one variable by solving the resulting
equation from step 3.
5. Find the value of the second variable by substituting the value
found in step 4 into either of the original equations.
6. Check the proposed solution in the original equation.
Section 4.1 – Systems of Linear Equations in Two Variables
Example:
a)
b)
Solve the following systems using the elimination
method.
x + 3y = 2
−x + 2y = 3
3x + 4 y = 2
2x + 5y = −1
page 5
Section 4.1 – Systems of Linear Equations in Two Variables
c)
−x + 3y = 6
3x − 9y = 9
page 6